Sharp error terms for return time statistics under mixing conditions *

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Sharp error terms for return time statistics under mixing conditions *

Miguel Abadi, Nicolas Vergne

To cite this version:

Miguel Abadi, Nicolas Vergne. Sharp error terms for return time statistics under mixing conditions

*. Journal of Theoretical Probability, Springer, 2009. �hal-02337198�

Sharp error terms for return time statistics under mixing conditions ^∗

Miguel Abadi

Nicolas Vergne

Abstract

We describe the statistics of repetition times of a string of symbols in a stochastic process. We consider a stringA of length n and prove: 1) The time elapsed until the process starting withArepeatsA, denoted by τA, has a distribution which can be well approximated by a degenerated law at the origin and an exponential law. 2) The number of consecutive repetitions of A, denoted by SA, has a distribution which is approxi- mately a geometric law. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and allow to get also approximations for all the moments ofτA and SA. Our results hold for processes that verify theφ-mixing condition.

Keywords: Mixing, recurrence, rare event, return time, sojourn time.

Running head: Return times under mixing conditions.

1 INTRODUCTION

This paper describes the return time statistics of a string of symbols in a mixing stochastic process with a finite alphabet. Generally speaking, the study of the time elapsed until the first occurrence of a small probability event in dependent processes has a long history, see for instance [10] and the references therein. The typical result is:

n→∞lim IP(τA_n> t bn |µ0) =e^−t, (1.1) whereτ_An is the first time the process hits a given measurable setA_n,n∈IN and such that the measureIP(A_n) go to zero as n→ ∞,{bn}n∈IN is a suitable re-scaling sequence of positive numbers andµ₀ is a given initial condition.

∗Work done as part of the activities of the project Projeto Temtico-PRONEX“Stochas- tic behavior, critical phenomena and rhythmic patterns identification in natural languages”.

Pros. FAPESP 03/09930-9.

†IMECC, Universidade Estadual de Campinas, P¸ca S´ergio Buarque de Holanda 651 Cid. Univ. CP 6065, Cep. 13083-859, Campinas SP, Brazil. Phone +55-19 37886021 miguel@ime.unicamp.br

‡Universit´e d’Evry Val d’Essonne, D´epartementcMath´ematiques, Laboratoire Statistique et G´enome, 91 000 Evry, France. vergne@genopole.cnrs.fr

Recently an exhaustive analysis of these statistics was motivated by ap- plications in different areas as entropy estimation, genome analysis, computer science, linguistic, among others. From the point of view of applications, a fun- damental task is to understand the rate of convergence of the limit (1.1). A detailed review of such results appearing in the literature can be found in [3].

It is the purpose of this paper to present the following new results: Forany stringAof lenght n

- A sharp upper bound for the above rate of convergence in general φ-mixing processes that holds whenµ₀=A.

- A sharp upper bound for the difference between the law of the number of consecutivevisits toA and a geometric law.

Whenµ₀ is taken asA, we refer to the distribution IP(τ_A > t | A) as the return time. In general it can not be well approximated by an exponential law.

This was firstly noted by Hirata, when he proved the convergence of the number of visits to a small cylinder around a point to the Poisson law. His result holds for axiom A diffeomorphisms (see [11]). The result holds foralmost every point.

Then, he proved that for periodic points, the asymptotic limit law of the return time differs from the one-level Poisson law, namelye^−t.

Our first result concerns the rate of convergence of limit in (1.1) whenµ0=A for any stringAof lenght n.. We prove that the return time law converges to a convex combination of a Dirac law at the origin and an exponential law.

Specifically, we show that for largen IP

τ_A> t IP(A) |A

≈

1 t≤IP(A)τ(A)

ζ_Ae^−ζAt t > IP(A)τ(A) .

τ(A) is the position of the first overlap ofAwith a copy of itself (see definition below). ζA is a parameter related to the overlap properties of the string A.

It is worth noting that the parameter of the exponential law is exactly the weight of the convex combination. So far, the overlap properties of a string appears as a major factor to describe the statistical properties of the return time. For instance, if a string overlaps itself, then it will turn out in the sequel that ζA 6= 1 and the return time distribution approximates the above mixture of laws. However, for a word which does not overlap itself, it will turn out that ζ_A= 1 and the return time distribution approximates a purely exponential law.

For the role of overlaps an a treatment of the independent case with a good introduction to the previous literature see [5], and for the Markov case with a probability generating functions point of view see [16].

It is worth recalling at this point that when in equation (1.1) the initial condition is the equilibrium measure of the process,τ_Ais called thehitting time ofA. In [12] it is proved a rate of convergence of the return time as function of the distance between the hitting time and return time laws. While this result applies only for cylinders around non-periodic points, our result applies toall of them.

The great enhancement of our work is that, contrarily to all the previous

decays exponentially fast int for all t >0. As a byproduct we obtain explicit expressions for all the moments of the return time. This also appears as a generalization of the famous Kac’s lemma (see [13]) which states that thefirst moment of the return time to a stringAof positive measure is equal toIP(A)⁻¹ and the result in [7] which presents conditions for theexistence of the moments of return times. Further, [12] proves that hitting and return times coincide if and only if the return time converges to the exponential law. We extend this result establishing that the laws of hitting and return times coincide if and only if the weight of the Dirac measure in the convex combination of the return time law is zero, which is equivalent to consider a non-overlapping string.

Our framework is the class of φ-mixing processes. For instance, irreducible and aperiodic finite state Markov chains are known to beψ-mixing (and then φ-mixing ) with exponential decay. Moreover, Gibbs states which have sum- mable variations areψ-mixing (see [17]). They have exponential decay if they have H¨older continuous potential (see [6]). However, sometimes the ψ-mixing condition is very restricted hypothesis difficult to test. We establish our result under the more general φ-mixing condition. Further examples of φ-mixing processes can be found in [14]. The error term is explicitly expressed as a func- tion of the mixing rateφ. We refer the reader to [9] for a source of examples and definitions of the several kinds of mixing processes.

The base of our proof is a sharp upper bound on the rate of convergence of the hitting time to an exponential law proved in [2].

The self-repeating phenomena in the distribution of the return time leads us to consider the problem of the sojourn time. Our second result states that the law of the number of consecutive repetitions of the string A, denoted by S_A, converges to a geometric law. Namely

IP(S_A=k |A)≈(1−ρ(A))ρ(A)^k . (1.2) Again here, the parameterρ(A) depends on the overlap properties of the string.

Furthermore we show that under suitable conditions one hasρ(A)≈1−ζA. As far as we know, this is the first result on this subject for dependent processes.

As in our previous result, the error bound we obtain decreases geometrically fast in k (see (1.2)). This decay on the error bound allows us to obtain an approximation forall the momentsofSAfor those of a geometrically distributed random variable.

Our results are applied in a forthcoming paper: In [4] the authors prove large deviations and fluctuations properties of the repetition time function introduced by Wyner and Ziv in [18] and further by Ornstein and Weiss in [15], and get entropy estimators.

This paper is organized as follows. In section 2 we establish our framework.

In section 3 we describe the self-repeating properties needed to state the return time result. In section 4 we establish the approximation for the return time

law. This is Theorem 4.1. Finally, in section 5 we state and prove the geometric approximation for the consecutive repetitions of a string. This is Theorem 5.1.

2 FRAMEWORK AND NOTATION

LetC be a finite set. Put Ω =C^Z. For each x= (xm)_m∈Z ∈Ω and m∈Z, letXm : Ω → C be the m-th coordinate projection, that is Xm(x) =xm. We denote byT : Ω→Ω the one-step-left shift operator, namely (T(x))m=xm+1. We denote by F the σ-algebra over Ω generated by strings. Moreover we denote byFI theσ-algebra generated by strings with coordinates inI,I⊆Z.

For a subsetA⊆Ω,A∈ Cⁿ if and only if

A={X0=a₀;. . .;X_n−1=a_n−1} , witha_i∈ C, i= 0, . . . , n−1.

We consider an invariant probability measureIP overF. We shall assume without loss of generality that there is no singleton of probability 0.

For two measurable setsV andW, we denote as usualIP(V|W) =IPW(V) = IP(V;W)/IP(W) the conditional measure ofV givenW. We writeIP(V;W) = IP(V ∩W).

We say that the process {Xm}_m∈ZZ is φ-mixing if the sequence φ(l) = sup|IPB(C)−IP(C)| ,

converges to zero. The supremum is taken over B and C such that B ∈ F_{0,.,n}, n∈IN, IP(B)>0, C ∈ F_{m≥n+l+1}.

We use the measure theoretic notation: {X_n^m=x^m_n}={Xn=x_n, . . . , X_m= x_m}. For an n-string A ={X₀ⁿ⁻¹ =xⁿ⁻¹₀ } and 1 ≤w ≤n, we write A^(w) = {X_n−wⁿ⁻¹ =xⁿ⁻¹_n−w}for thew-string belonging to theσ-algebra F{n−w,...,n−1}and consisting of thelastwsymbols ofA. We writeV^c= Ω\V, for the complement ofV.

The conditional mean of a r.v. X with respect to any measurable set V will be denoted by IEV(X) and we put IE(X) when V = Ω. Wherever it is not ambiguous we will writeCfor different positive constants even in the same sequence of equalities/inequalities. For brevity we put (a∨b) = max{a, b}and (a∧b) = min{a, b}.

3 PERIODS

Definition 3.1 Let A∈ Cⁿ. We define theperiodof A (with respect to T) as the numberτ(A)defined as follows:

τ(A) = min

k∈ {1, . . . , n} |A∩ T^−k(A)6=∅ .

By definition, if A∈ Cⁿ, thenA= (a0, . . . , a_n−1), ai∈ C for 0≤i≤n−1.

For instance, pick upA = (aaaabbaaaabbaaaa)∈ C¹⁵. Then shift a copy of A until there is a fit between them. Namely

A= aaaabb aaaabbaaa

T⁻⁶(A) = aaaabbaaa abbaaa .

Let us take A ∈ Cⁿ, and write n = q τ(A) +r, with q = [n/τ(A)] and 0≤r < τ(A). Thus

A=n

X₀^τ(A)−1=X_τ(A)^2τ(A)−1=. . .=X_(q−1)τ(A)^qτ(A)−1 =a^τ(A)−1₀ ; X_qτⁿ⁻¹_(A)=a^r−1₀ o . So, we say thatA hasperiod τ(A) andrest r. We remark that periods can be

“read backward” (and for the purpose of section 5 it will be more useful to do it in this way), that is

A=n

X₀^r−1=aⁿ⁻¹_n−r;Xn−(q−1)τ(A)−1

n−qτ(A) =..=X_n−2τ(A)^{n−τ(A)−1}=X_n−τ(A)ⁿ⁻¹ =aⁿ⁻¹_n−τ(A)o

=

(q−1)τ(A)

\

j=1

T^jτ(A)(A^(τ(A))) ∩ T^qτ(A)(A^(r)).

We recall the definition ofA^(w),1≤w≤n, at the end of section 2. For instance, using the previously chosenA,

A= (

period

z }| { aaaabb

period

z }| { aaaabb

rest

z}|{aaa) = (

rest

z }| { aaa

|{z}

T¹²A⁽³⁾ period

z }| { abbaaa

| {z }

T⁶A⁽⁶⁾ period

z }| { abbaaa

| {z }

A⁽⁶⁾

). (3.1)

In the middle of the above equality, periods are read forward while in the right hand side periods are read backward.

Consider the set of overlapping positions ofA:

k∈ {1, . . . , n−1} |A∩ T^−k(A)6=∅ ={τ(A), . . . ,[n/τ(A)]τ(A)} ∪ R(A), where

R(A) =

k∈ {[n/τ(A)]τ(A) + 1, . . . , n−1} |A∩ T^−k(A)6=∅ . The set{τ(A), . . . ,[n/τ(A)]τ(A)}is called the set of principal periods ofAwhile R(A) is called the set of secondary periods ofA. Furthermore, putr_A= #R(A).

Observe that one has 0≤r_A< n/2.

The notion of period is related to the notion ofretun times.

Definition 3.2 Given A∈ Cⁿ, we define the hitting time τA: Ω→IN∪ {∞}

as the following random variable: For anyx∈Ω τA(x) = inf{k≥1 :T^k(x)∈A} .

The return timeis the hitting time restricted to the set A, namely τA|A.

We remark the difference betweenτAandτ(A): whileτA(x) is the first time Aappears inx,τ(A) is the first overlapping position ofA.

Return times beforeτ(A) are not possible, thus,IPA(τA< τ(A)) = 0. Still, ifAdoes not return at timeτ(A), then it can not return at times kτ(A), with 2≤k≤[n/τ(A)], so one has

IP_A(τ(A)< τ_A≤[n/τ(A)]τ(A)) = 0.

The first possible return time afterτ(A) is nA=

minR(A) R(A)6=∅

n R(A) =∅ .

Furthermore, by definition ofR(A) one hasATT^−j(A) =∅ for allj such that [n/τ(A)]τ(A)< j≤n−1 andj6∈ R(A). Thus

IPA({[n/τ(A)]τ(A) + 1≤τA≤n−1} ∩ {τA6∈ R(A)}) = 0.

We finally remark that

T⁻ⁱA∩T^−jA=∅ ∀i, j∈ R(A).

Otherwise it would contradict the fact that the first return time toA is τ(A) since fori, j∈ R(A) one has|i−j|< τ(A). We conclude that

IP_A T⁻ⁱA∩T^−jA|i, j∈ R(A)

= 0. (3.2)

4 RETURN TIMES

ForA∈ Cⁿ define

ζ_A^def= IP_A(τ_A6=τ(A)) =IP_A(τ_A> τ(A)).

The equality follows by the comment at the end of the previous section.

It would be useful for the reader to note now that according to the comments of the previous section, one has

τA|A ∈ {τ(A)} ∪ R(A)∪ {k∈IN | k≥n}. (4.1) We now introduce the error terms that appear in the statement of our main result of this section.

Definition 4.1 Let us define

(A)^def= inf

0≤w≤nA

h

(2n+τ(A))IP(A^(w)) +φ(nA−w)i

. (4.2)

Theorem 4.1 Let {Xm}_m∈ZZ be a φ-mixing process. Then, for allA∈ Cⁿ, n∈ IN the following inequality holds for allt:

IPA(τA> t)−11{t<τ(A)}−11_{t≥τ(A)}ζAe^−ζAIP(A)(t−τ(A))

≤54(A)f(A, t), (4.3) wheref(A, t) =IP(A)te^−(ζA−16(A))IP(A)t.

We postpone an example showing the sharpness of(A) after Lemma 4.2.

Remark 4.1 A^(nA) is the part of the string A which does not overlap itself in A∩T^−nAA. Note that nA is the position of the first possible return time after τ(A). Recall that r_A = #R(A) and n_A = n if R(A) = ∅. Thus A^(w) with 1 ≤ w ≤ n_A is the part of the string A^(nA) after taking out its first n_A−w letters (this will be to create a gap of lengthn_A−wto use the mixing property).

Remark 4.2 WhenR(A) =∅, namely,Adoes not have secondary periods, the error(A)of Theorem 4.1 becomesinf_0≤w≤n

nIP(A^(w)) +φ(n−w) .

Remark 4.3 In the error term of the theorem, (A) provides a bound which shows the convergence uniform intof the return time law to that mixture of laws as the length of the string growths. The factor IP(A)t provides an extra bound for values of t smaller than 1/IP(A). The factor e^−(ζA−16(A)))IP(A)t provides an extra bound for values oftlarger than 1/IP(A).

Remark 4.4 On one hand IP(A) ≤Ce^−cn (see [1]). On the other hand, by construction nA > n/2. Further φ(n) → 0 as n → ∞. Taking for instance w=n/4 in (4.2) we warrant the smallness of(A)for large enough n.

Corollary 4.1 Let the process {X_m}_m∈ZZ be φ-mixing. Let β >0. Then, for allA∈ Cⁿ, n∈IN, the β-moment of the re-scaled timeIP(A)τ_A approaches, as n→ ∞, toΓ(β+ 1)/ζ_A^β−1. Moreover

IP(A)^βIEA(τ_A^β)−Γ(β+ 1) ζ_A^β−1

≤^∗(A)Cβ e2(A)(β+1)/ζ_A

ζ_A²

Γ(β+ 1)

ζ_A^β−1 , (4.4) where ^∗(A) = ((A)∨(nIP(A))^β), C > 0 is a constant and Γ is the analytic gamma function.

Remark 4.5 In particular, the corollary establishes that all the moments of the return time are finite.

Remark 4.6 In the special case when β = 1, the above corollary establishes a weak version of Kac’s Lemma (see [13]).

Remark 4.7 For each β fixed and nlarge enough one has β e^2(A)(β+1)/ζ_A² is close toβ/ζ_A². Thus in virtue of inequality (4.4), the corollary reads not just as a difference result but also as a ratio result.

The next corollary extends Theorem 2.1 in [12].

Corollary 4.2 Let the process {Xm}_m∈ZZ beφ-mixing. There exists a constant C >0 such that, for all A ∈ Cⁿ, n∈IN and allt >0 the following conditions are equivalent:

(a)

IP_A(τ_A> t)−e^−IP(A)t

≤C (A)f(A, t) , (b) |IP_A(τ_A> t)−IP(τ_A> t)| ≤C (A) f(A, t), (c)

IP(τ_A> t)−e^−IP(A)t

≤C (A)f(A, t), (d) |ζA−1| ≤C (A) .

Moreover, if{An}_n∈IN is a sequence of strings such thatIP(An)→0asn→ ∞, then the following conditions are equivalent:

(˜a) the return time law ofAn converges to a parameter one exponential law, (˜b) the return time law and the hitting time law ofAn converge to the same law, (˜c) the hitting time law ofAn converges to a parameter one exponential law, (d) The sequence˜ (ζA_n)n∈IN converges to one.

4.1 Preparatory results

Here we collect a number of results that will be useful for the proof of The- orem 4.1. In what follows and for shorthand notation we putfA= 1/(2IP(A)) (factor 2 is rather technical). The next lemma is a useful way to use the φ- mixing property.

Lemma 4.1 Let {Xm}_m∈ZZ be a φ-mixing process. Suppose that A ⊇ B ∈ F_{0,...,b}, C ∈ F_{{x∈IN|x≥b+n}} withb, g∈IN. The following inequality holds:

IPA(B;C)≤IPA(B) (IP(C) +φ(n)) .

Proof SinceB ⊆A, obviouslyIP(A∩B∩C) =IP(B∩C). By the φ-mixing property IP(B;C) ≤ IP(B) (IP(C) +φ(n)). Dividing the above inequality by IP(A) the lemma follows.

The following lemma says that return times overR(A) have small probabil- ity.

Lemma 4.2 Let {X_m}_m∈ZZ be a φ-mixing process. For all A ∈ Cⁿ, the fol- lowing inequality holds:

IPA(τA∈ R(A))≤(A). (4.5)

Proof For anywsuch that 1≤w≤nA

IPA(τA∈ R(A)) ≤ IPA



 [

j∈R(A)

T^−jA





≤ IPA



 [

j∈R(A)

T^−jA^(w)





≤ rAIP A^(w)

+φ(nA−w). (4.6) The first inequality follows by (3.2). SinceT^−jA⊂T^−jA^(w), second one follows.

Third one follows by the above lemma withB =AandC=∪_j∈R(A)T^−jA^(w). This ends the proof sincewis arbitrary.

Example 4.1 Consider a process {Xm}_m∈ZZdefined on the alphabetC={a, b}.

Consider the string introduced in (3.1):

A={(X0...X14) = (aaaabbaaaabbaaa)} .

Then,n= 15,τ(A) = 6,R(A) ={13,14},rA= 2andnA= 13. Thus A⁽¹³⁾={(X2...X14) = (aabbaaaabbaaa)} .

The φ-mixing property factorizes the probability

IPA





14

[

j=13

T^−jA



=IPA





14

[

j=13

T^−jA⁽¹³⁾



≤IPA





14

[

j=13

T^−jA^(w)



 . In such case, a gap att= 15of lengthwwith0≤w≤13is the best we can do to apply the φ-mixing property.

The next lemma will be used to get the non-uniform factor f(A, t) in the error term of Theorem 4.1.

Lemma 4.3 Let {Xm}_m∈ZZ be a φ-mixing process. Let A ∈ Cⁿ and let B ∈ F_{{x∈IN|x≥kfA}}, with k∈IN. Then the following inequality holds:

IPA(τA> kfA ; B)≤[IP(τA> fA−2n) +φ(n)]^k−1[IP(B) +φ(n)] . Proof First introduce a gap of length 2n between{τA > kfA} and B. Then use Lemma 4.1 to get the inequalities

IPA(τA> kfA ; B) ≤ IPA(τA> kfA−2n; B)

≤ IPA(τA> kfA−2n) [IP(B) +φ(n)] . (4.7)

Apply this procedure to{τA>(k−1)fA}andB=

τA◦T^(k−1)fA> fA−2n to boundIPA(τA> kfA−2n) by

IPA(τA>(k−1)fA−2n) [IP(τA> fA−2n) +φ(n)] . Iterate this procedure to boundIP_A(τ_A> kf_A−2n) by

IP_A(τ_A> f_A−2n) [IP(τ_A> f_A−2n) +φ(n)]^k−1 . This ends the proof of the Lemma.

The next proposition establishes a relationship between hitting and return times with an erroruniformint. In particular, (b) says that they are close (up to 2(A)) if and only ifζ_A is close to 1.

Proposition 4.1 Let {X_m}_m∈ZZ be a φ-mixing processes. LetA∈ Cⁿ andk a positive integer. Then the following holds:

(a) For all0≤r≤fA,

|IPA(τA> kfA+r)−IPA(τA> kfA)IP(τA> r)|

≤ 2(A)IP_A(τ_A> kf_A−2n) . (b) For alli≥τ(A)∈IN,

|IPA(τA> i)−ζAIP(τA> i)| ≤2(A). (4.8) Proof To simplify notation, for t∈Z we writeτ_A^[t] to mean τA◦T^t. Assume r ≥ 2n We introduce a gap of length 2n after coordinate t to construct the following triangular inequality

|IPA(τA> kfA+r)−IPA(τA> kfA)IP(τA> r)|

≤ |IPA(τA> kfA+r)−IPA(τA> kfA;τ_A^[kfA+2n] > r−2n)| (4.9) + |IPA(τA> kfA;τ_A^[kfA+2n]> r−2n)−IPA(τA> kfA)IP(τA> r−2n)|

(4.10) + IPA(τA> kfA)|IP(τA> r−2n)−IP(τA> r)|. (4.11) (4.9) is bounded by a direct computation by IP_A(τ_A > kf_A;τ_A^[kfA] ≤ 2n).

This last quantity is bounded using (4.7) by

IP_A(τ_A> kf_A−2n) [2nIP(A) +φ(n)] . Term (4.10) is bounded using the φ-mixing property by

IPA(τA> kfA)φ(n).

The modulus in (4.11) is bounded using stationarity by IP(τ_A≤2n)≤2nIP(A).

Ifr <2n, just changer−2nby zero and the same proof holds. This ends the proof of (a).

The proof of (b) is very similar to that previous one. We do it briefly. Write the following triangle inequality

|IPA(τA> i)−ζAIP(τA> i)|

≤ |IPA(τA> i)−IPA(τA> τ(A);τ_A^[τ(A)+2n]> i−τ(A)−2n)|

+ |IPA(τA> τ(A);τ_A^[τ(A)+2n]> i−τ(A)−2n)−ζAIP(τA> i−τ(A)−2n)|

+ ζ_A|IP(τ_A> i−τ(A)−2n)−IP(τ_A> i)|.

The moduli on the right hand side of the above inequality are bounded as follows.

The first one byIPA(τA > τ(A);τ_A^[τ(A)] ≤τ(A) + 2n−1) which is bounded by IPA(τA∈ R(A)∪ {n, . . . , τ(A) + 2n−1}).The cardinal ofR(A)∪ {n, . . . , τA+ 2n−1}is less or equal thann+τ(A) +R(A). Therefore, the last expression is bounded following the proof of Lemma 4.2 by (2n+τ(A))IP(A^(w)) +φ(nA−w).

The second one is bounded using the φ-mixing property byζAφ(n). The third one is bounded using stationarity by

IP(τ_A≤τ(A) + 2n)≤(τ(A) + 2n)IP(A).

This ends the proof of (b).

The following proposition is the key of the proof of Theorem 4.1.

Proposition 4.2 Let {Xm}_m∈ZZ be a φ-mixing process. Let A∈ Cⁿ, n∈IN and letkbe any integer k≥1. Then the following inequality holds:

IP_A(τ_A> kf_A)−IP_A(τ_A> f_A)IP(τ_A> f_A)^k−1

≤ 2(A)(k−1)IPA(τA> fA−2n)[IP(τA> fA−2n) +φ(n)]^k−2 . Proof Fork= 1 there is nothing to prove. Take k≥2. The left hand side of the above inequality is bounded by

k

X

j=2

|IPA(τA> jfA)−IPA(τA>(j−1)fA)IP(τA> fA)|IP(τA> fA)^k−j. The modulus in the above sum is bounded by

2(A)IP_A(τ_A>(j−1)f_A−2n),

due to Proposition 4.1 (a). The right-most factor is bounded using Lemma 4.3 by [IP(τA> fA−2n) +φ(n)]^j−2.The conclusion follows.

4.2 Proofs of Theorem 4.1 and corollaries

Proof of Theorem 4.1 We divide the proof according to the different values oft: (i)t < τ(A), (ii)τ(A)≤t≤f_Aand (ii)t > f_A.

Consider firstt < τ(A). (4.1) says that the left hand side of (4.3) is zero.

Consider nowτ(A)≤t≤fA. First write IP_A(τ_A> t) = IPA(τA> t)

IP(τ_A> t) IP(τ_A> t) =p_t+1IP(τ_A> t), (4.12) and

IP(τA> t) =

t

Y

i=τ(A)+1

IP(τA> i|τA> i−1) (4.13)

=

t

Y

i=τ(A)+1

(1−IP(T⁻ⁱA|τ_A> i−1))

=

t

Y

i=τ(A)+1

(1−piIP(A)), where

pi

def= IPA(τA> i−1) IP(τA> i−1) . Further

1−piIP(A)−e^−ζAIP(A)

≤ |pi−ζA|IP(A) +

1−ζAIP(A)−e^−ζAIP(A) . (4.14) Firstly, by Proposition 4.1 (b) and the fact thatIP(τA > i) ≥ 1/2 since i ≤ fA= 1/(2IP(A)) we have

|p_i−ζ_A| ≤ 2(A)

IP(τA> i) ≤4(A). (4.15) Secondly, note that |1−x−e^−x| ≤ x²/2 for all 0 ≤ x ≤ 1. Apply it with x=ζAIP(A) to bound the most right term of (4.14) by (ζAIP(A))²/2. Collecting the last two bounds we get

|1−piIP(A)−e^−ζAIP(A)| ≤ 9

2(A)IP(A), ∀i=τ(A) + 1, . . . , fA . Furthermore, since

|Y

ai−Y

bi| ≤max|ai−bi|(#i) max{ai;bi}^#i−1 ∀0≤ai, bi ≤1, (4.16)

we conclude from (4.13) and (4.12) that

|IP(τA> t)−e^−ζAIP(A)(t−τ(A))| ≤ 9

2(A)IP(A)t , (4.17) and

|IPA(τA> t)−ζAe^−ζAIP(A)(t−τ(A))| ≤ 9

2(A)IP(A)t , (4.18) for allτA≤t≤fA. This concludes this case.

Consider nowt > f_A. Write it as t=kf_A+rwithk a positive integer and 0≤r < f_A. We make the following triangle inequality

|IPA(τA> t)−ζAe^−ζAIP(A)(t−τ(A))|

≤ |IPA(τ_A> kf_A+r)−IP_A(τ_A> kf_A)IP(τ_A> r)| (4.19) + |IPA(τA> kfA)−IPA(τA> fA)IP(τA> fA)^k−1|IP(τA> r) (4.20) + |IP_A(τ_A> f_A)IP(τ_A> f_A)^k−1−ζ_Ae^−ζAk/2|IP(τ_A> r) (4.21) + ζAe^−ζAk/2 |IP(τA> r)−e^−ζAIP(A)(r−τ(A))| (4.22) By Proposition 4.1 (a), the modulus in (4.19) is bounded by

2(A)IPA(τA> kfA), and by Lemma 4.3

2(A)IP_A(τ_A> kf_A−2n)≤2(A)(IP(τ_A> f_A−2n) +φ(n))^k−1 . The modulus in (4.20) is bounded using Proposition 4.2 by

2(A)(k−1)(IP(τA> fA−2n) +φ(n))^k−2 . Thus, the sum of (4.19) and (4.20) is bounded by

2(A)(IP(τA> fA−2n) +φ(n))^k−2[k+φ(n)]. (4.23) On one handk+φ(n)≤k+ 1≤2k. On the other hand, applying (4.17) with t=fA−2nwe get

|IP(τA> fA−2n)−e^−ζA/2+(2n+τ(A))IP(A)| ≤9 4(A). Furthermore, by the Mean Value Theorem (MVT) we get

|e^−ζA/2+(2n+τ(A))IP(A)

−e^−ζA/2| ≤(2n+τ(A))IP(A)e(2n+τ(A))IP(A). We conclude that for large enoughn

|IP(τ_A> f_A−2n) +φ(n)−e^−ζA/2| ≤4(A)

And therefore (4.23) is bounded by

4(A)k(e^−ζA/2+ 4(A))^k−2 . (4.24) A direct computation using Taylor’s expansion gives

e^−ζA/2≤e^−ζA/2+ 4(A)≤e^{−(ζA/2−8(A))} . Sincet= (k/2IP(A)) +rwe get

e^{−(ζA/2)(k−2)}=e^{−ζAIP(A)t+ζA(IP(A)r+1)}, which is bounded by

e^{−ζAIP(A)t+3/2} . Similarly

e^−(ζA/2−8(A))(k−2)=e^−(ζA−16(A))IP(A)t+(ζA−16(A))(IP(A)r+1), which for large enoughnis bounded by

e^{−(ζA−16(A))IP(A)t+3/2} . Thus (4.24) is bounded by

36(A)IP(A)te^{−(ζA−16(A))IP(A)t}

To bound (4.21) we proceed as follows. From (4.17) and (4.18) witht=f_A we get that

|IP(τA> fA)−e^−ζA/2|

≤ |IP(τA> fA)−e^{−ζAIP(A)(fA−τ(A))}|+e^−ζA/2|e^{ζAIP(A)τ(A)}−1|

≤ 9

4(A) +nIP(A)

≤ 3(A), and similarly

|IP_A(τ_A> f_A)−ζ_Ae^−ζA/2| ≤3(A).

Applying the last two inequalities together with (4.16), we get that the modulus in (4.21) is bounded by

3(A)kmax{IP_A(τ_A> f_A);IP(τ_A> f_A);e^−ζA/2}^k−1

≤ 3(A)k

e^−ζA/2+ 3(A)k−1

.

An argument similar to that used to bound (4.24) let us conclude that the last expression is bounded by

The modulus in (4.22) is bounded using again (4.17) when r ≥ τ(A) by (9/2)(A). Ifr < τ(A) then it can be rewritten as

e^−ζAIP(A)(r−τ(A))−1 +IP(τA≤r),

which is bounded by 2nIP(A). We conclude that (4.22) is bounded by (9/2)(A)e^−ζAk/2= (9/2)(A)e^{−ζAIP(A)(t−r)}≤8(A)IP(A)te^−ζAIP(A)t. This ends the proof of the theorem.

Proof of Corollary 4.1 LetY be the r.v. with distribution given by P(Y > t) =

1 IP(A)< t≤IP(A)τ(A)

ζAe^{−ζA(t−IP(A)τ(A))} t < IP(A)τ(A) . Then we can rewrite (4.3) as

|IPA(IP(A)τA> t)−IP(IP(A)τA> t)| ≤C1(A)f(A, t/IP(A)). (4.25) Integrating (4.25) we get

IEA (IP(A)τA)^β

−IE Y^β

=

Z ∞ IP(A)

βt^β−1(IP(IP(A)τA> t)−IP(Y > t))

≤ Z ∞

IP(A)

βt^β−1|IP(IP(A)τA> t)−IP(Y > t)|

≤ C1(A) Z ∞

IP(A)

βt^β−1f(A, t/IP(A))dt . Now we compute IE Y^β

=R∞

IP(A)βt^β−1IP(Y > t). We do it in each interval [IP(A), IP(A)τ(A)] and [IP(A)τ(A),∞).

The first one is (IP(A)τ(A))^β−IP(A)^β. The second one can be re-written as

ζA e^ζAIP(A)τA Z ∞

0

−

Z IP(A)τ_A 0

!

βt^β−1e^−ζAtdt . (4.26) Consider the exponent of the second factor in (4.26). By definition we have ζAIP(A)τA ≤ IP(A)n. Moreover, IP(A) decays exponentially fast on n. Then for the second factor we have |e^ζAIP(A)τA−1| ≤ CIP(A)n. Further, the first integral is Γ(β+ 1)/ζ_A^β. The second one is bounded by (IP(A)τA)^β. We recall that the first factor in (4.26) isζA. We conclude that

IE Y^β

−Γ(β+ 1) ζ_A^β−1

≤CnIP(A) + 2(nIP(A))^β)≤C(nIP(A))^(β∧1).

Similar computations give Z ∞

IP(A)

βt^β−1f(A, t/IP(A))dt ≤ β β+ 1

Γ(β+ 2) (ζA−(A))^β+1

≤ βe2(A)(β+1)/ζ_A

ζ_A²

Γ(β+ 1) ζ_A^β−1 .

In the last inequality we usedx≤2(1−e^−x) for small enoughx >0. This ends the proof of the corollary.

Proof of Corollary 4.2. (a)⇔(d). It follows directly from Theorem 4.1.

(b)⇒(a),(c). It follows by Theorem 4.1 and Theorem 1 in [2]

(a)⇒(b) and (c)⇒(b). They follow by Theorem 4.1, Theorem 1 in [2] and (4.15). The corollary is proved.

5 SOJOURN TIME

In this section we consider the number of consecutive visits to a fixed string Aand prove that the distribution law of this number can be well approximated by a geometric law.

Definition 5.1 Let A ∈ Cⁿ. We define the sojourn time on the set A as the r.v. SA: Ω→IN∪ {∞}

S_A(x) = supn

k∈IN |x∈A∩T^−jτ(A)A ; ∀j= 1, . . . , ko , andSA(x) = 0if the supremum is taken over the empty set.

Before to state our main result we have to introduce the following defini- tion about certain continuity property of the probability IP conditioned to i consecutive occurrences of the stringA.

Definition 5.2 For each fixed A ∈ Cⁿ, we define the sequence of probabilities (p_i(A))_i∈IN as follows:

ρi(A)^def= IP



A

i

\

j=1

T^jτ(A)A



 . If the limitlim_n→∞ρ_i(A)exists then we denote it byρ(A).

Remark 5.1 By stationarity ρ1(A) = 1−ζA.

In the following 2 examples, the sequence (ρi(A))_i∈IN not just converges but

Example 5.1 For a i.i.d. Bernoulli process with parameter 0 < θ=IP(Xi = 1) = 1−IP(Xi = 0), and for the n-string A = {X₀ⁿ⁻¹ = 1}, we have that ρi(A) = 1−ζA=θ for alli∈IN.

Example 5.2 Let {Xm}_m∈ZZ be a irreducible and aperiodic finite state Markov chain. For A = {X₀ⁿ⁻¹ = aⁿ⁻¹₀ } ∈ Cⁿ, the sequence (ρ_i(A))_i∈IN is constant.

More precisely, by the Markovian property and for alli∈IN ρi(A) = IP

X_n−τ(A)ⁿ⁻¹ =aⁿ⁻¹_n−τ(A)|X_{n−τ(A)−1}=a_{n−τ(A)−1}

=

n−1

Y

j=n−τ(A)

IP(X_j=a_j|Xj−1=a_j−1) .

The next is an example of a process with infinity memory and converging (ρi(A))_n∈IN.

Example 5.3 The following is a family of processes of the renewal type. Define (Xn)n∈IN as the order one Markov chain over IN with transitions probabilities given by

Q(n, n+ 1) =qn Q(n,0) = 1−qn ∀n≥0 Define the process

Yn=

0 X_n= 0 1 X_n6= 0

The process (X_n)_n∈IN is positive recurrent (and then (Y_n)_n∈IN) if and only if P∞

k=0

Qk

j=0q_j <∞. Direct computations show that P(Y₀ⁿ⁻¹= 1) =

n

X

k=0 k

Y

j=0

qj ∀n∈IN .

Now chose qj such that P(Y₀ⁿ⁻¹ = 1) = e^−n+δ(n) with δ(n) any converging sequence (to any real number) and such that|δ(i+ 1)−δ(i)|<1for alli∈IN. TakeA={Y₀ⁿ⁻¹= 1}. Thusτ(A) = 1. Then

ρi(A) =e−1+δ(n+i+1)−δ(n+i) and lim

i→∞ρi(A) =e⁻¹∈(0,1) . In the following theorem we assume that (ρi(A))_i∈IN converges with velocity di(A). Namely, there is a real numberρ(A)∈[0,1) such that

|ρi(A)−ρ(A)| ≤di(A) for alli∈IN, (5.1) where d_i is a sequence converging to zero. For simplicity we put d(A) = sup{di(A)|i∈IN}.

Theorem 5.1 Let {Xm}_m∈ZZ be a stationary process. Let A ∈ Cⁿ. Assume that (5.1) holds. Then, there isc(A)∈[0,1), such that the following inequalities hold for allk∈IN:

IP_A(S_A=k)−(1−ρ(A))ρ(A)^k

≤c(A)^k

k+1

X

i=1

d_i(A)≤c(A)^k(k+ 1)d(A). We deduce immediately that theβ-moments ofSA can be approximated by IE(Y^β) whereY is a geometric random variable with parameterρ(A).

Corollary 5.1 LetY be a r.v. with geometric distribution with parameterρ(A).

Letβ >0. Then IEA

S_A^β

−IE(Y^β)

≤2d(A)

∞

X

k=1

k^β+1c(A)^k . Remark 5.2 The sumP∞

k=1k^β+1c(A)^k can be approximated using the Gamma function by Γ(β + 2)/(−lnc(A))^β+2. When the supremum of the distances

|ρ_i(A)−ρ(A)| is small, the approximations given by Theorem 5.1 and Corol- lary 5.1 are good. The smaller is c(A), the better they are. We compute these quantities for the examples of this section.

Example 5.1 (continuation) It follows straight-forward from definitions that ρi(A) = ρ(A) = IP(A^(τ(A))) for all i and for any A ∈ Cⁿ, n ∈ IN. Thus c(A) =IP(A^(τ(A))) andd(A) = 0.

Example 5.2(continuation) We already compute thatρi(A) =ρ(A) for alli and for anyA∈ Cⁿ, n∈IN. Thusc(A) =ρ(A) andd(A) = 0.

Example 5.3(continuation) For the samen-string there considered, we have d_i(A) =e⁻¹|eδ(n+i+1)−δ(n+i)−1| ≤ |δ(n+i+ 1)−δ(n+i)|,

and

d(A)≤sup{|δ(n+i+ 1)−δ(n+i)|, i∈IN} . So, for large enoughn, d(A) is small. Finally,

c(A) = sup{e⁻¹;e−1+δ(n+i+1)−δ(n+i), n∈IN} ∈(0,1). In the proof of Theorem 5.1 we will use the following lemma.

Lemma 5.1 Let(l_i)_i∈IN be a sequence of real numbers such that0≤l_i<1, for all i∈IN. Let 0≤l < 1 be such that|l_i−l| ≤d_i for all i ∈IN with d_i →0.

Then, there is a constantc∈[0,1), such that the following inequalities hold for allk∈IN:

k

Y

i=1

li−l^k

≤c^k−1

k

X

i=1

di ≤k c^k−1d .

Proof

k

Y

i=1

li−l^k

=

k

Y

i=1

li−

k−1

Y

i=1

lil+

k−1

Y

i=1

lil−

k−2

Y

i=1

lil²+

k−2

Y

i=1

lil²−. . .−l^k

≤

k

X

i=1





k−i

Y

j=1

l_j



|lk−i+1−l|lⁱ⁻¹≤c^k−1

k

X

i=1

d_i

≤ k c^k−1d , wherec= sup{l;li, i∈IN}.

Proof of Theorem 5.1 Fork= 0, we just note thatIPA(SA= 0) = 1−ρ1(A) and|1−ρ1(A)−(1−ρ(A))| ≤d1(A). Supposek≥1. Therefore

IPA(SA=k)

= IP_A





k

\

j=0

T^−jτ(A)A ; T^{−(k+1)τ(A)}A^c





= IP



T^{−(k+1)τ(A)}A^c|

k

\

j=0

T^−jτ(A)A





k

Y

i=1

IP



T^−iτ(A)A|

i−1

\

j=0

T^−jτ(A)A





= (1−ρk+1(A))

k

Y

i=1

ρi(A).

Third equality follows by stationarity. Lemma 5.1 ends the proof of the theorem.

Proof of Corollary 5.1We use the inequality IE X^β

−IE Y^β ≤X

k≥0

k^β|IP(X=k)−IP(Y =k)| ,

which holds for any pair of positive r.v. X, Y. We apply the above inequality withX=SA andY geometrically distributed with parameterρ(A).

The exponential decay of the error term in Theorem 5.1 ends the proof of the corollary.

AcknowledgmentsThe authors are beneficiaries of a Capes-Cofecub grant.

We thank P. Ferrari and A. Galves for useful discussions. We kindly thank also two anonymous referees for their useful comments and suggestions to improve a previous version of this article.

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